Topics in Algebra (FA23)

Course Info:

  • Lectures: Monday, Wednesday, Friday, block 12 (12:50 - 1:55 pm ET)
  • x-period: Tuesday, block 12X (1:20 - 2:10 pm EDT); generally used for extra office hours in Kemeny 105
  • Room: Kemeny 105
  • Instructor: Andrew Hanlon
  • Instructor office: Kemeny 320
  • E-mail: andrew.hanlon@dartmouth.edu
  • Office hours: Tu 11am - 12pm, W 11:30 am - 12:30 pm, or by appointment
  • Prerequisites: Math 22 or 24.  If you are unsure about your preparation, please talk to the instructor!
  • Required Text: Charles C. Pinter, A Book of Abstract Algebra, 2nd edition. References in the schedule below are to this book.
  • Grading: Grades will be based on weekly homework (40%), an in-class midterm exam (25%), participation in solving in-class exercises (5%), and a final exam (30%). Your lowest homework score will be dropped.

Midterm:

Our midterm is in-class on Wednesday, October 11, 2023. Here are some expectations for the exam. Here is a Practice Midterm and Solutions.  Here is the Midterm and Solutions.

Final exam:

Our final exam is on Tuesday, November 21, 2023 in 105 Kemeny Hall. Here are some expectations for the final exam. Here is a Practice final exam and Solutions. Here is the Final and Solutions.

Homework:

Homework is due on Gradescope on the day indicated below before 11:59 pm ET. Please upload your solutions by following the instructions in this video.  Feel free to submit handwritten or work typeset in LaTeX (Overleaf is a great resource and tool for the latter). If you are skipping a problem, it is easier for the grader if you write "Skip" and upload that as your solution to the problem. You are more than welcome to work with others on the homework problems. However, the solutions should be written up on your own and reflect your understanding.

Due Date Homework Solutions
9/14 Homework 0
9/21 Homework 1, tex file HW1 Solutions
9/28 Homework 2, tex file  HW2 Solutions
10/5 Homework 3, tex file HW3 Solutions
10/19 Homework 4, tex file HW4 Solutions
10/26 Homework 5, tex file HW5 Solutions
11/2 Homework 6, tex file HW6 Solutions
11/13 Homework 7, tex file HW7 Solutions

Schedule:

The specific topics covered in each class are tentative and will be updated as the semester progresses. Moreover, the notes posted for each lecture may contain slightly more content than the lecture and may bleed a bit into the next lecture. Let me know if you find any errors.

Date Topic References
M 9/11 Why Abstract Algebra? Ch. 1, Notes
W 9/13 Sets and proof  Appendix A, Notes
F 9/15 Operations and groups Ch. 2, Notes
M 9/18 More on groups Ch. 3, Notes
W 9/20 Basic properties of groups Ch. 4, Notes
F 9/22 Subgroups Ch. 5, Notes
M 9/25 Functions Ch. 6, Notes
W 9/27 Permutation groups Ch. 7, Notes
F 9/29 Cycle decomposition of permutations Ch. 8, Notes
M 10/2 Alternating group, Dihedral group Ch. 8, Notes
W 10/4 Isomorphisms Ch. 9, Notes
F 10/6 Order of group elements, cyclic groups Ch. 10,11, Notes
M 10/9 Review Notes
W 10/11 Midterm
F 10/13 Partitions and equivalence relations Ch. 12, Notes
M 10/16  Counting cosets Ch. 13, Notes
W 10/18 Homomorphisms Ch. 14, Notes
F 10/20 Quotient groups, fundamental homomorphism theorem Ch. 15,16, Notes
M 10/23 Group actions Notes
W 10/25 Integers, basic properties of rings Appendix B, Ch. 17, Notes
F 10/27 Ideals and ring homomorphisms Ch. 18, Notes
M 10/30 Quotient rings Ch. 19, Notes
W 11/1 Integral domains Ch. 20, Notes
F 11/3 Integers and prime factorization Ch. 21,22, Notes
M 11/6 Polynomial rings Ch. 24, Notes
W 11/8 Factoring polynomials Ch. 25, Notes
F 11/10 Polynomial substitution Ch. 26, Notes
M 11/13 Review Notes

Course Catalogue Description:

This course will provide an introduction to fundamental algebraic structures, and may include significant applications. The majority of the course will consist of an introduction to the basic algebraic structures of groups and rings. Additional work will consist either of the development of further algebraic structures or applications of the previously developed theory to areas such as coding theory or crystallography.

As a result of the variable syllabus, this course may not serve as an adequate prerequisite for MATH 81. Students who contemplate taking MATH 81 should consider taking MATH 71 instead of this course.

Learning Outcomes:

By the end of this course, you should be able to:

  1. Understand of the basic structures of algebra: define terms, explain their significance, and apply them in context;
  2. Solve mathematical problems: utilize abstraction and think creatively; 
  3. Write clear mathematical proofs: recognize and construct mathematically rigorous arguments.

Additional Pages:

  1. Academic Honor Principle
  2. Expectations
  3. Student Accessibility Needs
  4. Mental Health
  5. Title IX
  6. Religious Observances
  7. Additional Support for your Learning

Additional Resources: 

  • A 3Blue1Brown video advertising group theory.

  • Previous Math 31 webpages are available on the math department webpage. These may be another good source of exercises and practice exams.

  • It is highly recommended to solve as many exercises from the textbook or other resources as you have time for. This will be more useful than rereading your class notes or textbook as it will often force you to go back to look at the material anyway. The only way to truly learn mathematics is through practice. Feel free to ask about your solutions.

  • R. Hammack, Book of Proof is a free online textbook that you may find useful to sharpen your proof-writing skills.

  • There are many other textbooks and online materials on abstract algebra that you can consult. If you have questions on whether a particular one might be useful, do not hesitate to reach out. Here are just three possibilities: